The cost function $Z$ is given by $Z = 4x + 6y$. It is to be minimized. The feasible region for this function $Z$ is the shaded region represented in the following figure. Then the minimum value of $Z$ is and occurs at the point:

$A$ company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type $A$ require $5 \text{ minutes}$ each for cutting and $10 \text{ minutes}$ each for assembling. Souvenirs of type $B$ require $8 \text{ minutes}$ each for cutting and $8 \text{ minutes}$ each for assembling. There are $3 \text{ hours } 20 \text{ minutes}$ available for cutting and $4 \text{ hours}$ for assembling. The profit is $Rs. 5$ each for type $A$ and $Rs. 6$ each for type $B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

For the linear programming problem, $x + 2y \leq 10$, $3x + y \leq 12$, $x, y \geq 0$, the maximum value of $z = 5x + 10y$ occurs at every point on the line segment joining the points..

The minimum value of $t = 7x + 3y$ subject to constraints $x + y < 5$,$x + y < 10$,$x > 0$,$y > 0$ is . . . . . .

The objective function $z = 4x + 5y$ subject to the constraints $2x + y \geq 7$,$2x + 3y \leq 15$,$y \leq 3$,$x \geq 0$,and $y \geq 0$ has a minimum value at which point?

$A$ wholesale merchant wants to start a cereal business with $Rs \ 24000$. Wheat costs $Rs \ 400$ per quintal and rice costs $Rs \ 600$ per quintal. He has a storage capacity of $200$ quintals of cereal. He earns a profit of $Rs \ 25$ per quintal on wheat and $Rs \ 40$ per quintal on rice. If he stores $x$ quintals of rice and $y$ quintals of wheat,then for maximum profit,the objective function is: